We lived in the States for 4 years before moving back to Europe and then very soon after we moved to HKG. Not knowing that we were going to move here, I stocked up big time because most things are way more expensive in Europe. Looking back, I didn't need everything because we moved here, but it still worked out.
I wish I would have brought more shoes, like Stride Rites. I find it hard to buy good quality shoes for a reasonable price. However, you can wear sandals for a very long time, so it's quite hard to figure out when they need the regular shoes (I think I've done OK) and have a few pairs of SR sandals left too. Clothing, yes maybe, I stocked up quite a bit for my boy since I still had lots from my girl, but clothing is quite easy to get, I live around the corner from Stanley Market, so lots of OshKosh, Gap, etc. Most brands like Little Tykes and Step 2 are also way more expensive here, so if you need some toys (especially outdoor cars / wagon / ride-on things are fun at the playground), I would maybe bring one or two. I miss the Costco wipes dearly, I had a 6 month supply or so, but it was finished before moving here. I also loved their string garbage bags (almost finished). It's just the right quality, big quantity for the price Costco offers, which is great (I know it sounds crazy). I miss PBK dearly (I brought towels with their names, the Anywhere chairs with their names, just fun stuff, bins with their names).
I mean basically everything can be bought here, it's just that I got used to certain brands and things, which I miss now. I got a lot of nice deals, outlet shopping, etc. in the States so for us it worked out nicely and it was worth bringing it. It's a lot of "fun stuff". I love the GAP bras for myself, their Body stuff, etc. Every time people come here and ask me where do you get that ... I have to answer "States".
But again, many great things can be bought here, and a lot of copies can be made. I can't really think of something else, most stuff sold in the States is for sale here, at least compared to Europe.
I'll let you know if I think of something else.